Chain rule for differentiation

This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement for two functions

Suppose $f$ and $g$ are functions such that $g$ is differentiable at a point $x=x_{0}$, and $f$ is differentiable at $g(x_{0})$. Then the composite $f\circ g$ is differentiable at $x_{0}$, and we have:

${\frac {d}{dx}}[f(g(x))]|_{x=x_{0}}=f'(g(x_{0}))g'(x_{0})$

In terms of general expressions:

${\frac {d}{dx}}[f(g(x))]=f'(g(x))g'(x)$

In point-free notation, we have:

$(f\circ g)'=(f'\circ g)\cdot g'$

where $\cdot$ denotes the pointwise product of functions.

Related rules

• Chain rule for higher derivatives
• Product rule for differentiation
• Product rule for higher derivatives
• Differentiation is linear
• Inverse function theorem (gives formula for derivative of inverse function).